Power Series Domain: A power series $\sum c_n (x-a)^n$ acts like a function, but it only "works" for certain $x$ values. Outside this domain, it shoots to infinity.
The Ratio Test for Radius: We force convergence by insisting $\lim |a_{n+1}/a_n| < 1$. Solving for $x$ gives us a radius $|x-a| < R$.
Endpoint Testing: The Ratio Test fails when limit $=1$. We must manually test the exact border values $x = a-R$ and $x = a+R$!
Tags
Power SeriesConvergenceIntervalEndpoints
POWER SERIES SUM
$f(x) = \sum_{n=1}^{\infty}$ \frac{(x-2)^n}{n}
DRAG DOT TO TEST X VALUES FOR CONVERGENCE
Select Power Series Form
Interactive X-Value Tester
Testing $x$ = 3.00
RATIO TEST EVALUATION
Limit L: |x-2|
At $x=$ 3.0 : L = 1.0
THEORETICAL BOUNDS
Radius $R =$1
Interval:[1, 3)
Target Status:CONVERGES
Endpoint Checks: $R=1$ with Center $c=2$ means it converges if $|x-2|<1 \rightarrow 1